% LMITERM Add a term to a given LMI % % LMITERM(TERMID,A,B,FLAG) adds one term to some LMI in the LMI system % currently specified (see SETLMIS). A term is either an outer factor, a % constant matrix, or a variable term A*X*B or A*X'*B where X is a % matrix variable. The first input TERMID specifies which LMI and block % the term belongs to. % % Note: Because the OFF-DIAGONAL blocks (i,j) and (j,i) are transposed of % **** one another, specify the term content of ONLY ONE of these two % blocks. % % Input: % TERMID 4-entry vector specifying the term location and nature % Which LMI? % TERMID(1) = +n -> left-hand side of the n-th LMI % TERMID(1) = -n -> right-hand side of the n-th LMI % Which block? % For outer factors, set TERMID(2:3) = [0 0]. % Otherwise, set TERMID(2:3) = [i j] if the term % belongs to the (i,j) block of the LMI % What type of term? % TERMID(4) = 0 -> constant term % TERMID(4) = X -> variable term A*X*B % TERMID(4) = -X -> variable term A*X'*B % where X is the variable identifier returned by LMIVAR % A value of the outer factor, constant term, or left % coefficient in variable terms A*X*B or A*X'*B % B right coefficient in variable terms A*X*B or A*X'*B % FLAG quick way of specifying the expression A*X*B+B'*X'*A' % in a DIAGONAL block. Set FLAG='s' to specify it % with only one LMITERM command % % See also SETLMIS, LMIVAR, GETLMIS, LMIEDIT, NEWLMI. % Authors: P. Gahinet and A. Nemirovski 3/95 % Copyright 1995-2004 The MathWorks, Inc. % $Revision: 1.1.6.1 $ function lmiterm(termID,A,B,flag) global GLZ_HEAD GLZ_LMIS GLZ_LMIV GLZ_LMIT GLZ_DATA if nargin < 2, error('usage: lmiterm(termid,A,B,flag)'); elseif length(termID)~=4, error('TERMID must be a vector of length 4'); elseif nargin == 2 & termID(2)~=0 & termID(4)~=0, error('Both A and B must be specified for variable terms'); elseif any(size(GLZ_HEAD)~=[10 1]), error('Use SETLMIS to initialize the LMI system'); elseif nargin == 2, B=1; end Mv=[A(:);B(:)]; if isstr(A) | isstr(B), error('A and B must be matrices'); elseif any(any([imag(Mv) Mv==NaN abs(Mv)==Inf])), error('Complex numbers, NaN, or Inf not allowed in A,B'); elseif isempty(GLZ_LMIV), error('The LMI variables are undefined'); end Mv=[]; nlmi=GLZ_HEAD(1); nvar=GLZ_HEAD(2); nterm=GLZ_HEAD(3); ldt=GLZ_HEAD(7); lmi=termID(1); abslmi=abs(lmi); if lmi>0, str='lhs of '; else str='rhs of '; end row=abs(termID(2)); col=abs(termID(3)); var=termID(4); if row==0, col=0; var=0; end if ~row & nargin > 2, error('There should be no B matrix for outer factors'); elseif ~var & nargin > 2, error('There should be no B matrix for constant terms'); end % localize the specified LMI in GLZ_LMIS % (col of GLZ_LMIS corresp. to the label "abs(lmi)" lmirk=[]; if ~isempty(GLZ_LMIS), lmirk=find(GLZ_LMIS(1:nlmi,1)==abslmi); end if isempty(lmirk), lmirk=nlmi+1; blckdims=0; outsize=0; insize=0; begt=0; else outsize=GLZ_LMIS(lmirk,2); insize=GLZ_LMIS(lmirk,3); blckdims=GLZ_LMIS(lmirk,7:6+GLZ_LMIS(lmirk,6)); begt=GLZ_LMIS(lmirk,4); end if ~begt, begt=nterm+1; end % beginning of related terms in GLZ_LMIT % localize the specified matrix var in GLZ_LMIV % (varrk = col of GLZ_LMIV corresp. to the label "var") if var, varrk=find(GLZ_LMIV(1,1:nvar)==abs(var)); if row>0 & isempty(varrk), error('The specified matrix variable is undefined'); end end % misc if var~=0, vartype=GLZ_LMIV(2,varrk); if var<0 & (vartype==1 | vartype==31), var=-var; end end if row==0 | var~=0, if isempty(A), A=1; end, if isempty(B), B=1; end elseif isempty(A), error(sprintf('LMI #%d: empty constant term not permitted',abslmi)); end if col>row, % need to reflect the request wrt the diagonal olrow=row; row=col; col=olrow; if var==0, % cte term A=A'; else % var. term AA=A'; A=B'; B=AA; % reflected term will be B' Xk' A' if var<=0, var=-var; elseif (vartype~=1 & vartype~=31), % non symmetric variable var=-var; end end elseif col==row & var<0, AA=A'; A=B'; B=AA; var=-var; % for terms AXB+B'X'A', always store AXB end %%%%%%%%%%% HANDLE OUTER FACTORS FIRST %%%%%%%%%%%%%%% if row==0, % A -> outer factor [rA,cA]=size(A); mpl=rA*cA; scalarA=(mpl==1); if nterm>0 & lmirk <= nlmi, % extract the terms related to this LMI from GLZ_LMIT rgt=GLZ_LMIS(lmirk,4):GLZ_LMIS(lmirk,5); if ~isempty(rgt), lmit=GLZ_LMIT(1:2,rgt); else lmit=[0;0]; end if ~isempty(find(lmit(1,:)==lmi & ~lmit(2,:))), error(sprintf(... [ str 'LMI #%d: outer factor already defined'],abslmi)); end end if mpl>1, % full o.f. if insize<=0, insize=rA; end % outsize<0 -> scalar of previously met if outsize<0, outsize=insize; elseif ~outsize, outsize=cA; end elseif insize<=0 & outsize>0, insize=outsize; elseif outsize<=0 & insize>0, outsize=insize; elseif outsize<=0 & insize<=0, insize=-1; outsize=-1; end % if outsize==0, % insize=rA; outsize=cA; % elseif outsize==1 & insize==1, % insize=rA; outsize=rA; % end if (~scalarA & (rA~=insize | cA~=outsize)) | ... (scalarA & insize~=outsize), error(sprintf(... 'LMI #%d: the outer factors have incompatible dimensions',abslmi)); end % update GLZ_LMIS nterm=nterm+1; laux=length(blckdims); lnew=2+mpl; lmistore([abslmi outsize insize begt nterm laux blckdims],1,lmirk); GLZ_HEAD(1)=max(lmirk,nlmi); % update GLZ_LMIT lmistore([lmi;row;0;0;ldt;lnew],2,nterm); GLZ_HEAD(3)=nterm; % update GLZ_DATA lmistore([rA;cA;ma2ve(A,2)],3,ldt); GLZ_HEAD(7)=ldt+lnew; return elseif max(max(abs(A)))==0 | max(max(abs(B)))==0, return, end %%%%%%%%%%%% HANDLE OTHER TYPES OF TERMS %%%%%%%%%%%%%%%%%%% % update block dimensions (GLZ_LMIS) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [rA,cA]=size(A); mpl=rA*cA; scalarA=(mpl==1); [rB,cB]=size(B); mpr=rB*cB; scalarB=(mpr==1); laux=max(row,col); if laux > length(blckdims), blckdims(1,laux)=0; end % get term dimensions if var==0, rT=rA; cT=cA; else vadims=GLZ_LMIV(5:6,varrk); if var>0, rX=vadims(1); cX=vadims(2); else rX=vadims(2); cX=vadims(1); end scalarX=all(vadims==[1;1]); if scalarA | scalarX, rAX=max(rA,rX); cAX=max(cA,cX); elseif cA==rX, rAX=rA; cAX=cX; else error(sprintf(... [str 'LMI #%d, block (%d,%d): incompatible dimensions in A*X*B'],... abslmi,row,col)); end if scalarB | (rAX==1 & cAX==1), rT=max(rAX,rB); cT=max(cAX,cB); elseif cAX==rB, rT=rAX; cT=cB; else error(sprintf(... [str 'LMI #%d, block (%d,%d): incompatible dimensions in A*X*B'],... abslmi,row,col)); end end % update blckdims if (rT==1 & cT==1), % scalar term added rb=blckdims(row); cb=blckdims(col); if (rb==0 & cb==0), aux=min(blckdims)-1; blckdims(row)=aux; blckdims(col)=aux; elseif rb==0, blckdims(row)=cb; elseif cb==0, blckdims(col)=rb; elseif cb<0, ind=find(blckdims==cb); blckdims(ind)=rb*ones(1,length(ind)); elseif rb<0, ind=find(blckdims==rb); blckdims(ind)=cb*ones(1,length(ind)); elseif rb~=cb, error(sprintf([str 'LMI #%d, block (%d,%d): ' ... 'scalar term incompatible with block dimensions'],abslmi,row,col)); end elseif row==col, if rT~=cT, error(sprintf([str 'LMI #%d, block (%d,%d): ' ... 'terms in diagonal blocks must be square'],abslmi,row,col)); end rb=blckdims(row); if rb<0, ind=find(blckdims==rb); blckdims(ind)=rT*ones(1,length(ind)); elseif rb==0, blckdims(row)=rT; elseif rb~=rT, error(sprintf([str ... 'LMI #%d, block (%d,%d): term dimensions incompatible with other \n' ... ' terms in row #%d or column #%d'],... abslmi,row,col,row,col)); end else rb=blckdims(row); if rb<0, ind=find(blckdims==rb); blckdims(ind)=rT*ones(1,length(ind)); elseif rb==0, blckdims(row)=rT; elseif rb~=rT, error(sprintf([str ... 'LMI #%d, block (%d,%d): term dimensions incompatible with \n' ... ' other terms in same row'],... abslmi,row,col)); end cb=blckdims(col); if cb<0, ind=find(blckdims==cb); blckdims(ind)=cT*ones(1,length(ind)); elseif cb==0, blckdims(col)=cT; elseif cb~=cT, error(sprintf([str ... 'LMI #%d, block (%d,%d): term dimensions incompatible with \n' ... ' other terms in same column'],... abslmi,row,col)); end end % store term description %%%%%%%%%%%%%%%%%%%%%%%%% if var==0, % A -> constant coefficient % check symmetry in diagonal blocks if row==col, if max(max(abs(A'-A))) > 1.0e-8*max(max(abs(A))), error(sprintf(['LMI #%d, block (%d,%d): constant terms in ' ... 'diagonal blocks must be symmetric'],abslmi,row,row)); else A=(A+A')/2; end end % search for other constant terms in the same block jct=[]; if nterm>0 & lmirk<=nlmi, % extract the terms related to this LMI from GLZ_LMIT rgt=GLZ_LMIS(lmirk,4):GLZ_LMIS(lmirk,5); if ~isempty(rgt), lmit=GLZ_LMIT(1:4,rgt); jct=rgt(find(lmit(1,:)==lmi & lmit(2,:)==row &... lmit(3,:)==col & ~lmit(4,:))); end end if ~isempty(jct), % update already existing constant term resetct(lmirk,[],jct,A); return end newlt=[lmi;row;col;0;ldt;2+mpl]; newdata=[rA;cA;ma2ve(A,2)]; else % var. term: A -> left coeff, B -> right coeff bool=0; % handle diagonal terms (symmetry check) if row==col, % DIAG. BLOCK if vartype==1 | vartype==31 | scalarX, % SYM. VAR. if scalarX, % scalar variable bool=isymm(A*B); else rT=max(rT,blckdims(row)); if scalarA, AA=A*eye(rT); else AA=A; end if scalarB, BB=B*eye(rT); else BB=B; end bool=isslfcjg(AA,BB'); end if bool==1 & nargin==4, A=2*A; % AXB+B'XA' stored as 2 AXA' elseif bool==0 & nargin~=4, disp(sprintf('Warning in (%d,%d) block of LMI #%d:',... row,row,abslmi)); disp(' Non symmetric term AXB stored as (AXB+B''XA'')/2.'); disp(' For safety, set FLAG=''s'' to specify AXB+B''XA'' in one shot'); A=A/2; end elseif ~scalarX & nargin~=4, % NONSYM VAR. disp(sprintf('Warning in (%d,%d) block of LMI #%d:',... row,row,abslmi)); disp(' Non symmetric term AXB stored as (AXB+B''X''A'')/2.'); disp(' For safety, set FLAG=''s'' to specify AXB+B''X''A'' in one shot'); A=A/2; end end newlt=[lmi;row;col;var;ldt;5+mpl+mpr]; newdata=[rA;cA;ma2ve(A,2);rB;cB;ma2ve(B,2);bool]; end % update GLZ_LMIS nterm=nterm+1; laux=length(blckdims); lmistore([abslmi outsize insize begt nterm laux blckdims],1,lmirk); GLZ_HEAD(1)=max(lmirk,nlmi); % update GLZ_LMIT lmistore(newlt,2,nterm); GLZ_HEAD(3)=nterm; % update GLZ_DATA lmistore(newdata,3,ldt); GLZ_HEAD(7)=ldt+length(newdata); % Originator: P. Gahinet (last modified: March 1995)